Kourovka Notebook Problem 18.108

Definitions and mathematical statement

Regard a finitely generated group $G$ as a discrete group. Let $\widehat G$ denote its unitary dual: the set of unitary equivalence classes of irreducible unitary representations, equipped with the Fell topology. Write $1_G$ for the trivial representation and set \[ \widehat G_{\mathrm{fin}}=\{[\pi]\in\widehat G: \pi(G)\text{ is finite}\}. \] Following the supplied source, $G$ has property $(\tau)$ if $[1_G]$ is isolated in $\widehat G_{\mathrm{fin}}$ with its subspace topology; equivalently, there is an open subset $U\subseteq\widehat G$ such that \[ U\cap\widehat G_{\mathrm{fin}}=\{[1_G]\}. \] A finite simple group means a nontrivial finite group whose only normal subgroups are the identity subgroup and the whole group; the source imposes no nonabelian restriction.

Question (A. Jaikin-Zapirain, 2014). Does there exist a finitely generated group $G$ with property $(\tau)$ such that \[ \text{for every finite simple group }F,\quad \text{there exists a surjective homomorphism }\varphi_F:G\twoheadrightarrow F? \] The same group $G$ must work for every such $F$.

Supplied motivation and qualifications. The record attributes to E. Breuillard, B. Green, M. Kassabov, A. Lubotzky, N. Nikolov, and T. Tao a theorem asserting that there exist $\epsilon>0$ and $k\in{\mathbb{N}}$ such that every finite simple group $F$ has a generating set $S\subseteq F$ with $|S|\leq k$ for which the Cayley graph of $F$ with respect to $S$ is an $\epsilon$-expander. The constants are uniform over $F$. The supplied record does not specify the expansion normalization or the convention for symmetrizing $S$; this assertion is retained as contextual motivation.

The record also attributes to M. Ershov, A. Jaikin-Zapirain, and M. Kassabov, Mem. Amer. Math. Soc., 1186 (2017), the existence of a group with Kazhdan's property $(T)$ that maps surjectively onto every simple group of Lie type of rank at least $2$. Here property $(T)$ means that the trivial representation is isolated in the full unitary dual. This contextual assertion is preserved as supplied: its wording does not explicitly specify finiteness of the Lie-type groups or the rank convention. These contextual assertions are supplied literature reports, not independently verified conclusions. The registry labels the main question open; that label is also supplied rather than independently verified.

Short English statement

Does one finitely generated group with property $(\tau)$ have every finite simple group as a quotient? Here property $(\tau)$ requires the trivial representation to be isolated among irreducible unitary representations with finite image. The question is attributed to A. Jaikin-Zapirain (2014); its supplied status is open.

Sources

• Local UnsolvedMath registry, permanent ID $701536$, exact source code catalogue entry, title Kourovka Notebook Problem 18.108. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry supplies the status open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original Problem 18.108, 18th issue (2014); the record also identifies the 21st issue edition (2026), arXiv identifier 1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. Proposed by A. Jaikin-Zapirain in 2014. This is the supplied primary-source attribution; no source was independently consulted.
• E. Breuillard, B. Green, M. Kassabov, A. Lubotzky, N. Nikolov, and T. Tao: authors credited in the supplied record for the uniform-expander theorem used as motivation. No title, publication details, or link is supplied for this reference.
• M. Ershov, A. Jaikin-Zapirain, and M. Kassabov, Mem. Amer. Math. Soc., 1186 (2017): supplied reference for the property $(T)$ quotient assertion. No title or link is supplied. The bibliographic designation $1186$ is retained without assigning it a volume or issue interpretation.